Robot manipulators trends and development 2010 Part 10 doc

RobotManipulators,TrendsandDevelopment352

Fig. 16. Virtual impedance model for the mobile platform


Fig. 17.Obstacle repulsion force

The magnitude F
obs
is chosen to be (Borenstein & Koren, 1991):
 
2
F = a - b d(t) - d
min
obs obs obs
where a
obs
and b
obs
are positive constants satisfying the
condition
2
a = b (d - d )
max
min
obs obs
, d
max
is the maximum distance between the robot
and the detected obstacle that causes a nonzero repulsive force, d

min
represents the
minimum distance accepted between the robot and the obstacle and d(t) is the distance
measured between the robot and the obstacle d
min
< d(t) < d
max
( Fig. 17). Note that the
bound d
max
characterizes the repulsion zone. Which is inside the region where the repulsion
force has a non-zero value. Desired interaction impedance is defined as the linear dynamic
relationship Z
d
= B
d
s + K
d
where B
d
and K
d
are positive constants simulating the damping
and the spring effects, respectively, involved in the robot obstacle interaction inside the
repulsion zone.

3.4 Simulation results
Simulations are conducted in order to show the performance of the proposed methodology.
The numerical example supposes the lengths of the arm are such that
a = 0.6, a = 0.4, a = 0.3

1 2 3
and the initial configuration of the mobile manipulator is such
that: ξ = (0.1, 0.1, π/6)
p
and
T
/4]/2,/4,[
a
q


. The end effector is supposed to track
the following straight-line trajectory


 
T
T
ttttttt 1.01,1.0,1.0)(),(),()(
*
3
*
2
*
1
*


;
Furthermore, we imposed the following additional tasks to the mobile platform

* * * *
ξ (t) =(x (t),
y
(t), (t)) = (t, t, π/4)
p


. Fig. 18 shows the stance of the whole system when the
end effector tracks the reference trajectory. The resulting trajectory of the end effector as
well as that of the mobile plat form is depicted in Fig. 19. Figures 20, 21, 22 and 23describe
the evolution of the angles of the arm and the orientation of the platform respectively. If the
robot finds an obstacle at less than
d = 1m
max
the impedance control is activated, and the
collision is avoided as it can be seen in Fig. 24.

-0.1
0
0.1
0.2
0.
-0.4
-0.2
0
0.2
0.4
0
0.5
1

1.5
x
y
z

Fig. 18. A 3D-view of the arm and the mobile platform evolutions in an obstacle free space.

The resulted trajectories of the arm as well as of the mobile plat form appear in Fig. 26. The
corresponding curves showing the evolution of the angles of the arm and the orientation of
the platform are depicted in Figs. 27, 28, 29 and 30 respectively.

TrajectoryGenerationforMobileManipulators 353

Fig. 16. Virtual impedance model for the mobile platform


Fig. 17.Obstacle repulsion force

The magnitude F
obs
is chosen to be (Borenstein & Koren, 1991):


2
F = a - b d(t) - d
min
obs obs obs
where a
obs
and b

obs
are positive constants satisfying the
condition
2
a = b (d - d )
max
min
obs obs
, d
max
is the maximum distance between the robot
and the detected obstacle that causes a nonzero repulsive force, d
min
represents the
minimum distance accepted between the robot and the obstacle and d(t) is the distance
measured between the robot and the obstacle d
min
< d(t) < d
max
( Fig. 17). Note that the
bound d
max
characterizes the repulsion zone. Which is inside the region where the repulsion
force has a non-zero value. Desired interaction impedance is defined as the linear dynamic
relationship Z
d
= B
d
s + K
d

where B
d
and K
d
are positive constants simulating the damping
and the spring effects, respectively, involved in the robot obstacle interaction inside the
repulsion zone.

3.4 Simulation results
Simulations are conducted in order to show the performance of the proposed methodology.
The numerical example supposes the lengths of the arm are such that
a = 0.6, a = 0.4, a = 0.3
1 2 3
and the initial configuration of the mobile manipulator is such
that: ξ = (0.1, 0.1, π/6)
p
and
T
/4]/2,/4,[
a
q


. The end effector is supposed to track
the following straight-line trajectory


 
T
T

ttttttt 1.01,1.0,1.0)(),(),()(
*
3
*
2
*
1
*


;
Furthermore, we imposed the following additional tasks to the mobile platform
* * * *
ξ (t) =(x (t),
y
(t), (t)) = (t, t, π/4)
p


. Fig. 18 shows the stance of the whole system when the
end effector tracks the reference trajectory. The resulting trajectory of the end effector as
well as that of the mobile plat form is depicted in Fig. 19. Figures 20, 21, 22 and 23describe
the evolution of the angles of the arm and the orientation of the platform respectively. If the
robot finds an obstacle at less than
d = 1m
max
the impedance control is activated, and the
collision is avoided as it can be seen in Fig. 24.

-0.1

0
0.1
0.2
0.
-0.4
-0.2
0
0.2
0.4
0
0.5
1
1.5
x
y
z

Fig. 18. A 3D-view of the arm and the mobile platform evolutions in an obstacle free space.

The resulted trajectories of the arm as well as of the mobile plat form appear in Fig. 26. The
corresponding curves showing the evolution of the angles of the arm and the orientation of
the platform are depicted in Figs. 27, 28, 29 and 30 respectively.

RobotManipulators,TrendsandDevelopment354
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.5
1
1.5
2

2.5
3
3.5
4
4.5
5
x-y plots of the end-effector and mobile platform
x (m)
y (m)
end-effector
mobile platform

Fig. 19. End–effector and mobile platform trajectories in the x-y plane with no obstacles.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
The orientation qa1 of the arm
time (sec)
qa1 (rad)

Fig. 20. Articulation q
a1
curve
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-2.5
-2.4
-2.3
-2.2
-2.1
-2
-1.9
-1.8
-1.7
-1.6
-1.5
The orientation qa2 of the arm
time (sec)
qa2 (rad)

Fig.21. Articulation q
a2
curve
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.5
1
1.5
2
2.5
3
3.5
The orientation qa3 of the arm
time (sec)
qa3 (rad)

Fig. 22. Articulation q
a3
curve

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
The orientation

of platform
time (sec)

(rad)

Fig. 23. Articulation

curve

-2
0
2
4
6
-1

0
1
2
3
4
5
6
0
0.5
1
1.5
x
y
obstacle

Fig. 24. A 3D-View of the arm and the platform evolutions in presence of obstacles
TrajectoryGenerationforMobileManipulators 355
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x-y plots of the end-effector and mobile platform

x (m)
y (m)
end-effector
mobile platform

Fig. 19. End–effector and mobile platform trajectories in the x-y plane with no obstacles.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
The orientation qa1 of the arm
time (sec)
qa1 (rad)

Fig. 20. Articulation q
a1
curve
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-2.5
-2.4
-2.3
-2.2
-2.1
-2
-1.9

-1.8
-1.7
-1.6
-1.5
The orientation qa2 of the arm
time (sec)
qa2 (rad)

Fig.21. Articulation q
a2
curve
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.5
1
1.5
2
2.5
3
3.5
The orientation qa3 of the arm
time (sec)
qa3 (rad)

Fig. 22. Articulation q
a3
curve

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.5
0.55

0.6
0.65
0.7
0.75
0.8
0.85
The orientation

of platform
time (sec)

(rad)

Fig. 23. Articulation

curve

-2
0
2
4
6
-1
0
1
2
3
4
5
6

0
0.5
1
1.5
x
y
obstacle

Fig. 24. A 3D-View of the arm and the platform evolutions in presence of obstacles
RobotManipulators,TrendsandDevelopment356
0 1 2 3 4 5 6
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x-y plots of the end-effector and mobile platform
x (m)
y (m)
end-effector
mibile platform

Fig. 25. End–effector and mobile platform trajectories in the x-y plane in presence of
obstacles

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-600
-500
-400
-300
-200
-100
0
100
The orientation qa1 of the arm
time (sec)
qa1 (rad)

Fig. 26. Evolution curve of the joint
1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-2.6
-2.4
-2.2
-2
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
The orientation qa2 of the arm
time (sec)

qa2 (rad)

Fig. 27. Evolution curve of the joint
2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.5
1
1.5
2
2.5
3
3.5
The orientation qa3 of the arm
time (sec)
qa3 (rad)

Fig. 28. Evolution curve of the joint
3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
The orientation


of platform
time (sec)

(rad)

Fig. 29. Articulation

curve

4. Conclusion

This work proposed two different methodologies to generating desired joint trajectories for
both holonomic and non-holonomic mobile manipulators given prespecified operational
tasks. The first part considers a non-holonomic platform where the generalized inverses in
the resolution of a redundant system are used. The additional degrees of freedom are
exploited to avoid unforeseen obstacles and joint limits. In the second part of the work a
holonomic platfrom is used. In this case, the trajectory is generated using a reactive
approach based on virtual impedance and additional tasks. When the robot task is about a
stationary point, the mobile manipulator showed a good tracking for the manipulator. As
perspective an estimate procedure must be conducted in order to estimate the contact forces
and the unknown holonomic mobile manipulator parameters driving the system Computer
TrajectoryGenerationforMobileManipulators 357
0 1 2 3 4 5 6
0
0.5
1
1.5
2
2.5

3
3.5
4
4.5
5
x-y plots of the end-effector and mobile platform
x (m)
y (m)
end-effector
mibile platform

Fig. 25. End–effector and mobile platform trajectories in the x-y plane in presence of
obstacles
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-600
-500
-400
-300
-200
-100
0
100
The orientation qa1 of the arm
time (sec)
qa1 (rad)

Fig. 26. Evolution curve of the joint
1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-2.6
-2.4
-2.2
-2
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
The orientation qa2 of the arm
time (sec)
qa2 (rad)

Fig. 27. Evolution curve of the joint
2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.5
1
1.5
2
2.5
3
3.5
The orientation qa3 of the arm
time (sec)
qa3 (rad)

Fig. 28. Evolution curve of the joint
3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
The orientation

of platform
time (sec)

(rad)

Fig. 29. Articulation

curve

4. Conclusion

This work proposed two different methodologies to generating desired joint trajectories for
both holonomic and non-holonomic mobile manipulators given prespecified operational
tasks. The first part considers a non-holonomic platform where the generalized inverses in
the resolution of a redundant system are used. The additional degrees of freedom are
exploited to avoid unforeseen obstacles and joint limits. In the second part of the work a

holonomic platfrom is used. In this case, the trajectory is generated using a reactive
approach based on virtual impedance and additional tasks. When the robot task is about a
stationary point, the mobile manipulator showed a good tracking for the manipulator. As
perspective an estimate procedure must be conducted in order to estimate the contact forces
and the unknown holonomic mobile manipulator parameters driving the system Computer
RobotManipulators,TrendsandDevelopment358
simulations have validated to show the effectiveness of the two approaches. The reference
values obtained by the two methods can be used as inputs to controllers for real mtion.

5. References

Khatib, O. (1986). Real-time obstacle avoidance for manipulators and mobile robots,
International Journal of Robotics Research, 5(1):90{98.
Sundar, S. & Shiller, Z. (1997). Optimal obstacle avoidance based on the Hamilton-Jacobi-
Bellman equation,
IEEE Trans.on Robotics and Automation, Vol. 13, pp. 305{310.
Laumond, J. P., Jacobs, P. E., Taix, M., and Murray, R. M. (1994). A motion planner for
nonholonomic mobile robots,
IEEE Trans. on Robotics and Automation, Vol. 10, pp.
577{593.
Reeds, J. A. and Shepp, R. A. (1990). Optimal paths for a car that goes both forward and
backward,
Pacific J. Math., vol. 145, pp. 367–393.
Murray, R. M.; Li, Z. & Sastry, S. S. (1994). A Mathematical Introduction to Robotic
Manipulation.
Boca Raton, FL: CRC Press.
Tilbury, D.; Murray, R. M. & Sastry, S. S. (1995). Trajectory generation for the n-trailer
problem using goursatnormal form,
IEEE Trans. Automat. Contr., vol. 40, pp. 802–
819, May 1995.

Abdessemed, F. Monacelli, E. & Benmahammed, K. (2008). Trajectory Generation In an
Alternated and a Coordinated Motion Control Modes of a Mobile Manipulator,
AMSE journal, Modelling, Measurements and Control B, Vol.77, No 1, pp 18-34.
Djebrani, S. Benali, A. & Abdessemed, F. (2009). Force-position control of a holonomic
mobile manipulator, 12 int. Conf. on Climbing & Walking Robotsand the support
technologis for Mobile Machines Bogazaci Univ. Garanti Culture Center (North
Campus).
Qu, Z.; Wang, J. & Plaisted, C. E. (2004). A New Analytical Solution to Mobile Robot
Trajectory Generation in the Presence of Moving Obstacles,
IEEE Tran. on Robotics,
Vol. 20, No. 6.
Kant, K. & Zucker, S. W. (1988). Planning collision free trajectories in time varying
environments: A two-level hierarchy,
in Proc. IEEE Int. Conf. Robotics and
Automation
, Raleigh, NC, pp. 1644–1649.
Murray, R. M. & Sastry, S. S. (1993). Nonholonomic motion planning: Steering using
sinusoids,
IEEE Trans. Automat. Contr., vol. 38, pp. 700–716.
Abdessemed, F.; Benmahammed, K. & Eric Monacelli (2004). A Fuzzy Based Reactive
Controller for Non-Holonomic Mobile Robot,
Journal of Robotics and Autonomous
Systms
, 47 (2004) 31-46.
Russell, S. & Norvig, P. (2000). Artificial Intelligence: A Modern Approach,
Prentice Hall,
New Jersey, 1995
A. Okabe, B. Boots, K. Sugihara and S.N. Chiu, Spatial Tessellations and Applications of
Voronoi Diagrams, John Wiley & Sons, New York.
Zhao, M.; Ansari, N. & Hou, E.S.H. (1994). Mobile manipulator path planning by a genetic

algorithm,
Journal of Robotic Systems, 11(3): 143-153.
Pin, F. G. & Culioli, J. C. (1992). Optimal Positioning of Combined Mobile Platform-
Manipulator systems for Material Handling Tasks,
Journal of intelligent and Robotic
Systems. 6: 165-182.
Pin, F. G.; Morgansen, K. A.; Tulloch, F. A.; Hacker, C. J. & Gower, K. B. (1996). Motion
Planning for Mobile Manipulators with a Non-Holonomic Constraint Using the FSP
(Full Space Parameterization) Method,
Journal of Robotic Systems 13(11), 723-736.
Lee, J. K. & Cho, H. S. (1997). Mobile manipulator Motion Planning for Multiple Tasks Using
Global Optimization Approach,
Journal of Intelligent and Robotic Systems, 18: 169-190.
Seraji, H. (1995) Configuration control of rover-mounted manipulators,
IEEE Int. Conf. on
Robotics and Automation, pp2261-2266.
Campion, G.; Bastin, B. & D'Andrea-Novel. (1996). Structural proprieties and classifcation of
kinematic and dynamic models of wheeled mobile robots.
IEEE Trans. on Robotics
and Automation
, 2(1):47{62, February.
Liegeois, A. (1997). Automatic supervisory control of the configuration and behavior of
multibody mechanisms,
IEEE Trans. Syst. Man Cybernet. 7, 842-868.
Seraji, H. (1993). An on-line approach to coordinated mobility and manipulation, ICRA’93,
pp. 28-35, May, 1993.
Mourioux, G.; Novales, C.; Poisson, G. & Vieyres, P. (2006). Omni-directional robot with
spherical orthogonal wheels: concepts and analyses,
IEEE International Conference on
Robotics and Automation, pp. 3374-3379.

Seraji, H. (1998). A unified approach to motion control of mobile manipulators,
The
International Journal of Robotics Research
, vol. 17, no. 2, pp. 107-118.
Bayle, B.; Fourquet, J. Y.; Lamiraux, F. & Renaud, M. (2002). Kinematic control of wheeled
mobile manipulators, IEEE/RSJ International Conference on Intelligent Robots and
Systems, pp. 1572-1577.
Arai, T. & Ota, J. (1996). Motion planning of multiple mobile robots using virtual
impedance,
Journal of Robotics and Mechatronics, vol. 8, no. 1, pp. 67-74.
Borenstein, J. & Koren, Y. (1991). The vector field histogram fast obstacle avoidance for
mobile robots,
IEEE Transactions on Robotics and Automation, vol. 7, no. 3, pp. 278-
288.
TrajectoryGenerationforMobileManipulators 359
simulations have validated to show the effectiveness of the two approaches. The reference
values obtained by the two methods can be used as inputs to controllers for real mtion.

5. References

Khatib, O. (1986). Real-time obstacle avoidance for manipulators and mobile robots,
International Journal of Robotics Research, 5(1):90{98.
Sundar, S. & Shiller, Z. (1997). Optimal obstacle avoidance based on the Hamilton-Jacobi-
Bellman equation,
IEEE Trans.on Robotics and Automation, Vol. 13, pp. 305{310.
Laumond, J. P., Jacobs, P. E., Taix, M., and Murray, R. M. (1994). A motion planner for
nonholonomic mobile robots,
IEEE Trans. on Robotics and Automation, Vol. 10, pp.
577{593.
Reeds, J. A. and Shepp, R. A. (1990). Optimal paths for a car that goes both forward and

backward,
Pacific J. Math., vol. 145, pp. 367–393.
Murray, R. M.; Li, Z. & Sastry, S. S. (1994). A Mathematical Introduction to Robotic
Manipulation.
Boca Raton, FL: CRC Press.
Tilbury, D.; Murray, R. M. & Sastry, S. S. (1995). Trajectory generation for the n-trailer
problem using goursatnormal form,
IEEE Trans. Automat. Contr., vol. 40, pp. 802–
819, May 1995.
Abdessemed, F. Monacelli, E. & Benmahammed, K. (2008). Trajectory Generation In an
Alternated and a Coordinated Motion Control Modes of a Mobile Manipulator,
AMSE journal, Modelling, Measurements and Control B, Vol.77, No 1, pp 18-34.
Djebrani, S. Benali, A. & Abdessemed, F. (2009). Force-position control of a holonomic
mobile manipulator, 12 int. Conf. on Climbing & Walking Robotsand the support
technologis for Mobile Machines Bogazaci Univ. Garanti Culture Center (North
Campus).
Qu, Z.; Wang, J. & Plaisted, C. E. (2004). A New Analytical Solution to Mobile Robot
Trajectory Generation in the Presence of Moving Obstacles,
IEEE Tran. on Robotics,
Vol. 20, No. 6.
Kant, K. & Zucker, S. W. (1988). Planning collision free trajectories in time varying
environments: A two-level hierarchy,
in Proc. IEEE Int. Conf. Robotics and
Automation
, Raleigh, NC, pp. 1644–1649.
Murray, R. M. & Sastry, S. S. (1993). Nonholonomic motion planning: Steering using
sinusoids,
IEEE Trans. Automat. Contr., vol. 38, pp. 700–716.
Abdessemed, F.; Benmahammed, K. & Eric Monacelli (2004). A Fuzzy Based Reactive
Controller for Non-Holonomic Mobile Robot,

Journal of Robotics and Autonomous
Systms
, 47 (2004) 31-46.
Russell, S. & Norvig, P. (2000). Artificial Intelligence: A Modern Approach,
Prentice Hall,
New Jersey, 1995
A. Okabe, B. Boots, K. Sugihara and S.N. Chiu, Spatial Tessellations and Applications of
Voronoi Diagrams, John Wiley & Sons, New York.
Zhao, M.; Ansari, N. & Hou, E.S.H. (1994). Mobile manipulator path planning by a genetic
algorithm,
Journal of Robotic Systems, 11(3): 143-153.
Pin, F. G. & Culioli, J. C. (1992). Optimal Positioning of Combined Mobile Platform-
Manipulator systems for Material Handling Tasks,
Journal of intelligent and Robotic
Systems. 6: 165-182.
Pin, F. G.; Morgansen, K. A.; Tulloch, F. A.; Hacker, C. J. & Gower, K. B. (1996). Motion
Planning for Mobile Manipulators with a Non-Holonomic Constraint Using the FSP
(Full Space Parameterization) Method,
Journal of Robotic Systems 13(11), 723-736.
Lee, J. K. & Cho, H. S. (1997). Mobile manipulator Motion Planning for Multiple Tasks Using
Global Optimization Approach,
Journal of Intelligent and Robotic Systems, 18: 169-190.
Seraji, H. (1995) Configuration control of rover-mounted manipulators,
IEEE Int. Conf. on
Robotics and Automation, pp2261-2266.
Campion, G.; Bastin, B. & D'Andrea-Novel. (1996). Structural proprieties and classifcation of
kinematic and dynamic models of wheeled mobile robots.
IEEE Trans. on Robotics
and Automation
, 2(1):47{62, February.

Liegeois, A. (1997). Automatic supervisory control of the configuration and behavior of
multibody mechanisms,
IEEE Trans. Syst. Man Cybernet. 7, 842-868.
Seraji, H. (1993). An on-line approach to coordinated mobility and manipulation, ICRA’93,
pp. 28-35, May, 1993.
Mourioux, G.; Novales, C.; Poisson, G. & Vieyres, P. (2006). Omni-directional robot with
spherical orthogonal wheels: concepts and analyses,
IEEE International Conference on
Robotics and Automation, pp. 3374-3379.
Seraji, H. (1998). A unified approach to motion control of mobile manipulators,
The
International Journal of Robotics Research
, vol. 17, no. 2, pp. 107-118.
Bayle, B.; Fourquet, J. Y.; Lamiraux, F. & Renaud, M. (2002). Kinematic control of wheeled
mobile manipulators, IEEE/RSJ International Conference on Intelligent Robots and
Systems, pp. 1572-1577.
Arai, T. & Ota, J. (1996). Motion planning of multiple mobile robots using virtual
impedance,
Journal of Robotics and Mechatronics, vol. 8, no. 1, pp. 67-74.
Borenstein, J. & Koren, Y. (1991). The vector field histogram fast obstacle avoidance for
mobile robots,
IEEE Transactions on Robotics and Automation, vol. 7, no. 3, pp. 278-
288.
RobotManipulators,TrendsandDevelopment360
TrajectoryControlofRobotManipulatorsUsingaNeuralNetworkController 361
Trajectory Control of Robot Manipulators Using a Neural Network
Controller
Zhao-HuiJiang
x

Trajectory Control of Robot Manipulators
Using a Neural Network Controller

Zhao-Hui Jiang
Hiroshima Institute of Technology
Japan

1. Introduction

Many advanced methods proposed for control of robot manipulators are based on the
dynamic models of the robot systems. Model-based control design needs a correct dynamic
model and precise parameters of the system. Practically speaking, however, every dynamic
model has some degrees of incorrectness and every parameter associates with some degrees
of identification error. The incorrectness and errors eventually result positioning or
trajectory tracking errors, and even cause the system to be unstable. In the past two decades,
intensive research activities have been devoted on the design of robust control systems and
adaptive control systems for the robot in order to overcome the control system drawback
caused by the model errors and uncertain parameters, and a great number of research
results have been reported, for example, (Hsia, 1989), (Kou, and Wang, 1989), (Slotine and Li,
1989), ( Spong, 1992), and (Cheah, Liu and Slotine, 2006). However, almost parts of results
associate with complicated control system design approaches and difficulties in the control
system implementation for industrial robot manipulators.
Recently, neural network technology attracts many attentions in the design of robot
controllers. It has been pointed out that multi-layered neural network can be used for the
approximation of any nonlinear function. Other advantages of the neural networks often
cited are parallel distributed structure, and learning ability. They make such the artificial
intelligent technology attractive not only in the application areas such as pattern recognition,
information and graphics processing, but also in intelligent control of nonlinear and
complicated systems such as robot manipulators (Sanger, 1994), (Kim and Lewis, 1999),
(Kwan and Lewis, 2000), (Jung and Yim, 2001) (Yu and Wang, 2001). A new field in robot

control using neural network technology is beginning to emerge to deal with the issues
related to the dynamics in the robot control design. A neural network based dynamics
compensation method has been proposed for trajectory control of a robot system (Jung and
Hsia, 1996). A combined approach of neural network and sliding mode technology for both
feedback linearization and control error compensation has been presented (Barambones and
Etxebarria, 2002). Sensitivity of a neural network performance to learning rate in robot
control has been investigated (Clark and Mills, 2000).

In the following, we present a simple control system consisting of a traditional controller
and a neural network controller with parallel structure for trajectory tracking control of
16
RobotManipulators,TrendsandDevelopment362

industrial robot manipulators. First, a PD controller is designed. Second, a neural network
with three layers is designed and added to the control system in the parallel way to the PD
controller. Finally, a learning scheme used to train the weights of each layer of the neural
network is derived by minimizing a criterion prescribed in a quadratic form of the error
between a planed trajectory and response of the robot. Control system implementation issue
is discussed. Both the motivation function of the neural network and dynamic model used
in the calculation of the learning law are simplified to meet practical needs. An industrial
manipulator AdeptOne is adopted as an experimental test bed. Trajectory tracking control
simulations and experiments are carried out. The results demonstrate effectiveness and
usefulness of the proposed control system.

2. Dynamic models of robot manipulators

2.1 Torque-based dynamic model
A torque-based dynamic model of robot manipulator describes relationship between motion
and joint torque of the robot without concerning what generates the torque and how. This
class of dynamic formulation is most popular and widely used in the control design and

simulation of the robot manipulator. Usually, a torque-based dynamic model can be
systematically derived by using the Lagrange method as follows

τθgθθθHθθM  )(),()(

(1)

where,
n
Rθ
and
n
Rτ
are joint variable and torque,
nn
R

)(θM
is inertia matrix,
n
RθθθH

),(
contains Coriolis and centrifugal forces, and
n
R)(θg
denotes
gravitational force.
Remarks: In motion equation (1),
)(θM

is a symmetric matrix, and
),(2)( θθHθM




is a
skew symmetric matrix. These properties of robot dynamics allow one to design the control
system on the basis of dynamic model in an easier way.

2.2 Voltage-based dynamics model
In the almost cases of industrial robot manipulators, the torque-based dynamic model
cannot be used directly because most industrial manipulators are not functionally designed
on the basis of torque/force control but servo control. In the other words, as actuators
almost all robot manipulators are equipped with servo motors that are controlled by input
voltage not by current. The former results the so-called velocity servo, and the later meets
the needs of the torque-based control that may require the torque-based dynamic model.
For, the robot with servo-controlled motors, we need to take the characteristics of the motors
and servo-units into consideration in the dynamic modeling, parameter identification and
control design. Generally, the dynamic model of the motor with a servo unit can be given as
follows.

iiiivii
i
i
i
i
u)θ(Dθfτ
a

R
τ
a
L



(2)


For the n-link robot manipulator, the dyanmic characteristics of the motors and servo
units can be rewritten in a compact form as

uθDθfτRατLα 

)(
11


v
(3)

Though rates of amplifiers of the servo units are included in the parameters in the above
equation, in the following, we rather like to use the nominal terms of parameters that are
often referred directly to a servo motor.
),,,(
21 n
uuudiag 

u

denotes the input voltage of
the servo units;
),,,(
21 n
LLLdiag L
,
),,,(
21 n
RRRdiag 

R
are matrices of inductance
and resistance; and
),,,(
21 n
diag





α
is the matrix with the elements being the back
electromotive constant of each servo motor;
),,,(
21 vnvvv
fffdiag 

f
denotes matrix of

viscous friction constants, and
)(θD

is a diagonal matrix that diagonal elements indicate the
constants of Coulomb frictions and electrical dead zones of the motors. Combining (1) and
(3) together, after some simple manipulations we obtain


uθθ,Hθθθ,RθθL  )(
ˆ
)(
ˆ
)(
ˆ

(4)
where
)()(
ˆ
1
θMLaθL


(5)



)(),()()(
ˆ
11

θMRaθθHθMLaθθ,R





(6)





)(),()(),()(
ˆ
11
θgθθθHRaθgθθθHLaθθ,H 






(7)

3. Control problem statement

Standing on the theoretical view point, the dynamic model given by (4) can be used in
model-based control system design with a kind of computed-torque like control method.
Implementation of such a control system, however, is difficult to carry out since either
acceleration sensors or numerical derivative approaches are necessary for calculating the

control input that contains acceleration feedback. Acceleration sensors are not available in
industrial robots, and the numerical derivative approaches would result high frequency
noises and phase lag. On the other hand, every dynamic model contains more or less
modeling errors and/or parameter uncertainties that cause imprecise trajectory tracking in
the control based on the dynamic model.
Although what we are discussing here is about high-performance advanced control
methods, the practical world that we have to face is that all commercialized industrial robot
manipulators associate with built-in traditional PID controllers. However, a significant
drawback of the PID control system is that it cannot guarantee a precise tracking result for
given dynamic trajectories since such the control system is essentially driven by trajectory
error itself.
From the above discussion, we clarified the problems in dynamic trajectory control of robot
manipulators, and found two key points for the problem-solving in the dynamic trajectory
tracking control system design: one is how to utilize the built-in PID controller of the robot
system; another one is how to take the dynamic characteristics of the robot into
TrajectoryControlofRobotManipulatorsUsingaNeuralNetworkController 363

industrial robot manipulators. First, a PD controller is designed. Second, a neural network
with three layers is designed and added to the control system in the parallel way to the PD
controller. Finally, a learning scheme used to train the weights of each layer of the neural
network is derived by minimizing a criterion prescribed in a quadratic form of the error
between a planed trajectory and response of the robot. Control system implementation issue
is discussed. Both the motivation function of the neural network and dynamic model used
in the calculation of the learning law are simplified to meet practical needs. An industrial
manipulator AdeptOne is adopted as an experimental test bed. Trajectory tracking control
simulations and experiments are carried out. The results demonstrate effectiveness and
usefulness of the proposed control system.

2. Dynamic models of robot manipulators

2.1 Torque-based dynamic model
A torque-based dynamic model of robot manipulator describes relationship between motion
and joint torque of the robot without concerning what generates the torque and how. This
class of dynamic formulation is most popular and widely used in the control design and
simulation of the robot manipulator. Usually, a torque-based dynamic model can be
systematically derived by using the Lagrange method as follows

τθgθθθHθθM  )(),()(

(1)

where,
n
Rθ
and
n
Rτ
are joint variable and torque,
nn
R

)(θM
is inertia matrix,
n
RθθθH

),(
contains Coriolis and centrifugal forces, and
n
R)(θg

denotes
gravitational force.
Remarks: In motion equation (1),
)(θM

is a symmetric matrix, and
),(2)( θθHθM




is a
skew symmetric matrix. These properties of robot dynamics allow one to design the control
system on the basis of dynamic model in an easier way.

2.2 Voltage-based dynamics model
In the almost cases of industrial robot manipulators, the torque-based dynamic model
cannot be used directly because most industrial manipulators are not functionally designed
on the basis of torque/force control but servo control. In the other words, as actuators
almost all robot manipulators are equipped with servo motors that are controlled by input
voltage not by current. The former results the so-called velocity servo, and the later meets
the needs of the torque-based control that may require the torque-based dynamic model.
For, the robot with servo-controlled motors, we need to take the characteristics of the motors
and servo-units into consideration in the dynamic modeling, parameter identification and
control design. Generally, the dynamic model of the motor with a servo unit can be given as
follows.

iiiivii
i
i

i
i
u)θ(Dθfτ
a
R
τ
a
L



(2)


For the n-link robot manipulator, the dyanmic characteristics of the motors and servo
units can be rewritten in a compact form as

uθDθfτRατLα 

)(
11


v
(3)

Though rates of amplifiers of the servo units are included in the parameters in the above
equation, in the following, we rather like to use the nominal terms of parameters that are
often referred directly to a servo motor.
),,,(

21 n
uuudiag 

u
denotes the input voltage of
the servo units;
),,,(
21 n
LLLdiag L
,
),,,(
21 n
RRRdiag R
are matrices of inductance
and resistance; and
),,,(
21 n
diag



α
is the matrix with the elements being the back
electromotive constant of each servo motor;
),,,(
21 vnvvv
fffdiag f
denotes matrix of
viscous friction constants, and
)(θD


is a diagonal matrix that diagonal elements indicate the
constants of Coulomb frictions and electrical dead zones of the motors. Combining (1) and
(3) together, after some simple manipulations we obtain


uθθ,Hθθθ,RθθL  )(
ˆ
)(
ˆ
)(
ˆ

(4)
where
)()(
ˆ
1
θMLaθL


(5)



)(),()()(
ˆ
11
θMRaθθHθMLaθθ,R






(6)





)(),()(),()(
ˆ
11
θgθθθHRaθgθθθHLaθθ,H 






(7)

3. Control problem statement

Standing on the theoretical view point, the dynamic model given by (4) can be used in
model-based control system design with a kind of computed-torque like control method.
Implementation of such a control system, however, is difficult to carry out since either
acceleration sensors or numerical derivative approaches are necessary for calculating the
control input that contains acceleration feedback. Acceleration sensors are not available in
industrial robots, and the numerical derivative approaches would result high frequency

noises and phase lag. On the other hand, every dynamic model contains more or less
modeling errors and/or parameter uncertainties that cause imprecise trajectory tracking in
the control based on the dynamic model.
Although what we are discussing here is about high-performance advanced control
methods, the practical world that we have to face is that all commercialized industrial robot
manipulators associate with built-in traditional PID controllers. However, a significant
drawback of the PID control system is that it cannot guarantee a precise tracking result for
given dynamic trajectories since such the control system is essentially driven by trajectory
error itself.
From the above discussion, we clarified the problems in dynamic trajectory control of robot
manipulators, and found two key points for the problem-solving in the dynamic trajectory
tracking control system design: one is how to utilize the built-in PID controller of the robot
system; another one is how to take the dynamic characteristics of the robot into
RobotManipulators,TrendsandDevelopment364

consideration in the trajectory control. The neural network provides us with some new
options in the control design in many ways: to approximate dynamics and/or inverse
dynamics, to compensate dynamic effects, to be a controller itself, etc. In this chapter, we
combine the built-in controller and a neural network together to design a new control
system for trajectory control of the robot. We aim at high precision trajectory tracking
control of the industrial robot manipulators using simple and applicable control method.
We design a control strategy with both technologies of PID control a neural network for
taking the advantages of both simplicity on design and implementation of a PID controller,
and learning capacity of neural network control. The main idea is to establish a control
system with the PID controller and a neural network control scheme which are parallel to
each other in structure for achieving precise tracking control of dynamic trajectories. The
detail description of the control system design yields to the next section.

4. Structures of the robot control system using neural network

It is usual that the neural network controllers are structurely degined as the feedback
controllers in the control system. The neural networks are trained such that the trjactory
tracking erorr
e converge to zero. Fig.1 and Fig.2 show two kinds of block structures of the


Fig. 1. Structure of a neural network control system with the trajectory error being the input
of the neural network.


Fig. 2. Structure of a neural network control system with the state variable being the input of
the neural network.

neural network system. In Fig.1, the neural network is driven by the trajectory trcking error,
whereas in Fig.2 the neural network is dirven by the states of the robot system.
+
-
u

Robot
Manipulator
Neural Network
Controller
θθ

,

dd
θθ


,


+
-

u

Robot
Manipulator
Neural Network
Controller
θθ

,

dd
θθ

,

e

e


To combine a neural network controller and a built-in PID controller together in parallel, we
have two ways according to two structures shown in Fig.1 and Fig.2. In detail, the control
system black diagrams are given in Fig. 3 and Fig.4.

Fig. 3. Structure of robot control system with the trajectory error being the input of the
neural network.


Fig. 4. Structure of robot control system with the state variable being the input of the neural
network.

In control system of Fig.3, what role the neural network controller plays is no more than a
controller since the neural network’s input is trajectory error. On the other hand, the neural
network controller given in Fig.4 has possibility to work as not only a controller but also a
dynamic compensator. The later generates the forces/torques to compensate the gravity and
other dynamic forces/torques according to the dynamic trajectories so that the trajectory
tracking may be more accurately achieved. In the rest part of this chapter, we will mainly
discuss the design of control system that the structure is shown in Fig.4.

5. Control system design

5.1 The control strategy
In the control system shown in Fig.4, the total control scheme is given as follows.

n
l
uuu 

(8)

l
u
is control input of the PID controller, and can be simply described as below.

+
-
u

+

Robot
Manipulator

PID Controller
Neural Network
Controller
+

θθ

,

l
u

n
u

dd
θθ

,

e

+
-

u

+

Robot
Manipulator

PID Controller
Neural Network
Controller
+

θθ

,

l
u

n
u

dd
θθ

,

e

TrajectoryControlofRobotManipulatorsUsingaNeuralNetworkController 365

consideration in the trajectory control. The neural network provides us with some new
options in the control design in many ways: to approximate dynamics and/or inverse
dynamics, to compensate dynamic effects, to be a controller itself, etc. In this chapter, we
combine the built-in controller and a neural network together to design a new control
system for trajectory control of the robot. We aim at high precision trajectory tracking
control of the industrial robot manipulators using simple and applicable control method.
We design a control strategy with both technologies of PID control a neural network for
taking the advantages of both simplicity on design and implementation of a PID controller,
and learning capacity of neural network control. The main idea is to establish a control
system with the PID controller and a neural network control scheme which are parallel to
each other in structure for achieving precise tracking control of dynamic trajectories. The
detail description of the control system design yields to the next section.

4. Structures of the robot control system using neural network

It is usual that the neural network controllers are structurely degined as the feedback
controllers in the control system. The neural networks are trained such that the trjactory
tracking erorr
e converge to zero. Fig.1 and Fig.2 show two kinds of block structures of the


Fig. 1. Structure of a neural network control system with the trajectory error being the input
of the neural network.

Fig. 2. Structure of a neural network control system with the state variable being the input of
the neural network.

neural network system. In Fig.1, the neural network is driven by the trajectory trcking error,
whereas in Fig.2 the neural network is dirven by the states of the robot system.
+
-
u

Robot
Manipulator
Neural Network
Controller
θθ

,

dd
θθ

,


+
-

u

Robot
Manipulator

Neural Network
Controller
θθ

,

dd
θθ

,

e

e


To combine a neural network controller and a built-in PID controller together in parallel, we
have two ways according to two structures shown in Fig.1 and Fig.2. In detail, the control
system black diagrams are given in Fig. 3 and Fig.4.


Fig. 3. Structure of robot control system with the trajectory error being the input of the
neural network.


Fig. 4. Structure of robot control system with the state variable being the input of the neural
network.

In control system of Fig.3, what role the neural network controller plays is no more than a
controller since the neural network’s input is trajectory error. On the other hand, the neural

network controller given in Fig.4 has possibility to work as not only a controller but also a
dynamic compensator. The later generates the forces/torques to compensate the gravity and
other dynamic forces/torques according to the dynamic trajectories so that the trajectory
tracking may be more accurately achieved. In the rest part of this chapter, we will mainly
discuss the design of control system that the structure is shown in Fig.4.

5. Control system design

5.1 The control strategy
In the control system shown in Fig.4, the total control scheme is given as follows.

n
l
uuu 
(8)

l
u
is control input of the PID controller, and can be simply described as below.
+
-

u

+

Robot
Manipulator

PID Controller

Neural Network
Controller
+

θθ

,

l
u

n
u

dd
θθ

,

e

+
-

u

+

Robot
Manipulator

PID Controller
Neural Network
Controller
+

θθ

,

l
u

n
u

dd
θθ

,

e

RobotManipulators,TrendsandDevelopment366

dt
t
d
i
d

p
d
v
l


0
)()()( θθkθθkθθku

(9)

where
d
θ
and
d
θ

are planned trajectories of joint displacements and velocities,
v
k
,
p
k
, and
i
k
are gain matrices.
n
u

is the control input of the neural network controller being designed. The structure of
neural network controller is shown in Fig. 5. The detail mathematical description of the
neural network is given by

f(Wq)Vu 
n
(10)

where
nT
nn
R
2
2121
],,,,,[ 




q
denotes input vector with elements being each
joint variable, velocity;
m
nmnnn
Ruuu  ],[
21
u
is output vector,
ln
R



2
W
and
ml
R

V
with their elements being expressed by
ij
w
and
jk
v
, are weight matrices from
input nodes to the hidden layer and from hidden layer to the output layer;
l
R)(f
is an
activation function vector of the hidden layer with elements being selected as a saturation
function, such as a sigmoid function; l is the number of hidden nodes. Though the
dimension of robot joint inputs equals joint numbers n, here we denote it as m in order to
describe the network controller design clearer.


Fig. 5. Multilayer neural network controller.

2.2 Detail design of the neural network controller
For tracking control of a robot with a designed dynamic trajectory, only the PD controller is

not enough to ensure a proper tracking precision. For this reason, we design the neural
network controller such that it takes the important part on which the PD controller has
shown its limitation and/or powerlessness. In doing so, the neural network controller
should be trained in such the way: the trajectory tracking error getting smaller and smaller
while training. First, we choose a performance criterion of the whole control system with a
quadratic form of the trajectory tracking error and velocity tracking error, as follows.
1
q

jk
v

1n
u

2
q

n
q
2

nm
u

ij
w






))((
2
1
)()()()(
qqqq
θθθθθθθθ


dd
TT
2
1
2
1
E

dddd
(11)

The weights’ learning algorithm is derived based on the back-propagation approach. The
tuning law is to give weights’ increments to be proportional to the negative gradient of the
performance criterion with respect to the weights. For updating of the weights between the
hidden layer and the output layer, we define an increment as

jk
jkjk
v
E

v




( j=1,2,…l; k=1,2,…m ) (12)

where, j and k indicate the one between jth node of the hidden layer and kth node of the
output layer , and
jk

is a constant of proportionality, to be designed as a learning rate.
Whereas for the weights between the input layer and hidden layer, we give

ij
ijij
w
E
w




( i=1,2…2n; j=1,2,…l ) (13)

where
ij

is a learning rate to be designed by the user.
Using the chain rule and noting that the weights are independent with

l
u
, the partial
derivative of (12) can be expressed as follows,

jk
nk
nkjk
v
u
u
E
v
E









q
q
(14)

In detail,
k
vE

2
/
and
jkk
uv


/
2
can be given as

nTT
d
R
E
21
)(




eqq
q
(15)

12 



n

k
nk
R
u
b
q
(16)

and
j
jk
f
v
u



nk
(17)

where
j
f
is the output of jth node of the hidden layer.
Similarly, one can use the chain rule to (13) to obtain
TrajectoryControlofRobotManipulatorsUsingaNeuralNetworkController 367

dt
t
d

i
d
p
d
v
l


0
)()()( θθkθθkθθku

(9)

where
d
θ
and
d
θ

are planned trajectories of joint displacements and velocities,
v
k
,
p
k
, and
i
k
are gain matrices.

n
u
is the control input of the neural network controller being designed. The structure of
neural network controller is shown in Fig. 5. The detail mathematical description of the
neural network is given by

f(Wq)Vu

n
(10)

where
nT
nn
R
2
2121
],,,,,[ 




q
denotes input vector with elements being each
joint variable, velocity;
m
nmnnn
Ruuu  ],[
21
u

is output vector,
ln
R


2
W
and
ml
R

V
with their elements being expressed by
ij
w
and
jk
v
, are weight matrices from
input nodes to the hidden layer and from hidden layer to the output layer;
l
R)(f
is an
activation function vector of the hidden layer with elements being selected as a saturation
function, such as a sigmoid function; l is the number of hidden nodes. Though the
dimension of robot joint inputs equals joint numbers n, here we denote it as m in order to
describe the network controller design clearer.


Fig. 5. Multilayer neural network controller.

2.2 Detail design of the neural network controller
For tracking control of a robot with a designed dynamic trajectory, only the PD controller is
not enough to ensure a proper tracking precision. For this reason, we design the neural
network controller such that it takes the important part on which the PD controller has
shown its limitation and/or powerlessness. In doing so, the neural network controller
should be trained in such the way: the trajectory tracking error getting smaller and smaller
while training. First, we choose a performance criterion of the whole control system with a
quadratic form of the trajectory tracking error and velocity tracking error, as follows.
1
q

jk
v

1n
u

2
q

n
q
2

nm
u

ij
w





))((
2
1
)()()()(
qqqq
θθθθθθθθ


dd
TT
2
1
2
1
E

dddd
(11)

The weights’ learning algorithm is derived based on the back-propagation approach. The
tuning law is to give weights’ increments to be proportional to the negative gradient of the
performance criterion with respect to the weights. For updating of the weights between the
hidden layer and the output layer, we define an increment as

jk

jkjk
v
E
v




( j=1,2,…l; k=1,2,…m ) (12)

where, j and k indicate the one between jth node of the hidden layer and kth node of the
output layer , and
jk

is a constant of proportionality, to be designed as a learning rate.
Whereas for the weights between the input layer and hidden layer, we give

ij
ijij
w
E
w




( i=1,2…2n; j=1,2,…l ) (13)

where
ij


is a learning rate to be designed by the user.
Using the chain rule and noting that the weights are independent with
l
u
, the partial
derivative of (12) can be expressed as follows,

jk
nk
nkjk
v
u
u
E
v
E









q
q
(14)

In detail,
k
vE
2
/
and
jkk
uv  /
2
can be given as

nTT
d
R
E
21
)(




eqq
q
(15)

12 



n

k
nk
R
u
b
q
(16)

and
j
jk
f
v
u



nk
(17)

where
j
f
is the output of jth node of the hidden layer.
Similarly, one can use the chain rule to (13) to obtain
RobotManipulators,TrendsandDevelopment368

ij
j
j

j
j
n
n
T
ij
w
z
z
f
f
E
w
E












 u
u
q
q

(18)

where
j
z
is the summation of input signals to jth node of the hidden layer, i.e.



n
s
ssjj
qwz
2
1
. In detail, each undetermined terms are given as

mn
m21
R




2
],,[ bbbb
u
q

n

(19)

m
j
T
jmj2j1
n
Rvvv
f



v
u
],,[ 
j
(20)

i
j
q
w
z



ij
(21)

In (19), b is a matrix depended on the dynamics of the robot, and will be specified in the next

subsection.
k
b
in (16) is the kth column vector of b.
The fourth partial derivative term of right side of (18) can be determined directly using
partial derivative
jjjjj
zzfzf  /)(/
for a designed activation function
)(
jj
zf
.

5.3 An implementation issue
In the design of the neural network controller, since we aimed at trajectory tracking
performance of the system, we designed the performance criterion using error’s quadratic
form of the inputs of the neural network other than using error’s quadratic form of the
outputs of the neural network, though the later is much usual in neural network design. It
eventually results the use of dynamics of the system in deriving the learning law with back
propagation method since the inputs and outputs of the robot system and neural network
controller are contrary to each other. Ignoring the small parameters, usually dynamics (3)
can be simplified as

uθθ,hθθL  )()(

(22)

Using
TTT

],[ θθq


as a state variable vector, above expression can be rewritten as

Bupq 

(23)

where











),()(
1
θθhθL
θ
p


(24)









)(
1
θL
0
B
(25)
Generally, the solution of (23) can be given by




tt
dd
00
Bupq
(26)

One can numerically calculate b in a real time control process as

BB
u

q
b








k
i
ii
t
ttd
1
1
0
)(

n
(27)

where
i
t indicates the ith sampling time.

6. Simulation and experimental studies

6.1 The test bed

The experimental test bed used in this research is an AdeptOne XL robot manipulator
shown in Fig.6. It is a SCARA type high performance Direct Drive (DD) industrial robot
manipulator possessed with 4 joints. Except the third joint being a prismatic joint, all joints
are revolute. Though a closed-loop servo system is built-in by Adept Technology
Corporation on the basis of servo units and servo motors, using the Advanced Servo Library
the user is allowed to access the D/A converter directly to establish a user-designed close-
loop servo system for the development of more advanced control system by V+ language.
We developed control software on such the software and hardware environment.
Since the third joint is prismatic and dynamically independent with other joints, control
subsystem for the third joint can be designed independently and easily.



Fig. 6. AdeptOne robot manipulator


TrajectoryControlofRobotManipulatorsUsingaNeuralNetworkController 369

ij
j
j
j
j
n
n
T
ij
w
z
z

f
f
E
w
E












 u
u
q
q
(18)

where
j
z
is the summation of input signals to jth node of the hidden layer, i.e.




n
s
ssjj
qwz
2
1
. In detail, each undetermined terms are given as

mn
m21
R




2
],,[ bbbb
u
q

n
(19)

m
j
T
jmj2j1
n
Rvvv
f




v
u
],,[ 
j
(20)

i
j
q
w
z



ij
(21)

In (19), b is a matrix depended on the dynamics of the robot, and will be specified in the next
subsection.
k
b
in (16) is the kth column vector of b.
The fourth partial derivative term of right side of (18) can be determined directly using
partial derivative
jjjjj
zzfzf






/)(/
for a designed activation function
)(
jj
zf
.

5.3 An implementation issue
In the design of the neural network controller, since we aimed at trajectory tracking
performance of the system, we designed the performance criterion using error’s quadratic
form of the inputs of the neural network other than using error’s quadratic form of the
outputs of the neural network, though the later is much usual in neural network design. It
eventually results the use of dynamics of the system in deriving the learning law with back
propagation method since the inputs and outputs of the robot system and neural network
controller are contrary to each other. Ignoring the small parameters, usually dynamics (3)
can be simplified as

uθθ,hθθL  )()(

(22)

Using
TTT
],[ θθq



as a state variable vector, above expression can be rewritten as

Bupq



(23)

where











),()(
1
θθhθL
θ
p


(24)









)(
1
θL
0
B
(25)
Generally, the solution of (23) can be given by




tt
dd
00
Bupq
(26)

One can numerically calculate b in a real time control process as

BB
u
q

b








k
i
ii
t
ttd
1
1
0
)(

n
(27)

where
i
t indicates the ith sampling time.

6. Simulation and experimental studies

6.1 The test bed
The experimental test bed used in this research is an AdeptOne XL robot manipulator

shown in Fig.6. It is a SCARA type high performance Direct Drive (DD) industrial robot
manipulator possessed with 4 joints. Except the third joint being a prismatic joint, all joints
are revolute. Though a closed-loop servo system is built-in by Adept Technology
Corporation on the basis of servo units and servo motors, using the Advanced Servo Library
the user is allowed to access the D/A converter directly to establish a user-designed close-
loop servo system for the development of more advanced control system by V+ language.
We developed control software on such the software and hardware environment.
Since the third joint is prismatic and dynamically independent with other joints, control
subsystem for the third joint can be designed independently and easily.



Fig. 6. AdeptOne robot manipulator


RobotManipulators,TrendsandDevelopment370

Focusing on control of the most complex part of the robot, we do not take the third joint into
consideration in the control design. The fourth joint is extremely light-weight designed
comparing with other joints and its link length is zero. Fourth joint does not cause dynamic
coupling to the others. Therefore, we do not take it into consideration as well in control
design and experiments.

6.2 Trajectory tracking control simulations
The joint trajectory tracking control simulations were carried out based on a simplified
dynamic model of (3). The neural network controller was designed with three layers, four
nodes for the input layer and hidden layer respectively, and two nodes for the output layer.
The learning scheme was designed using the method given in section IV. The desired joint
trajectories are designed using triangle functions with amplitudes to be 45 and 30 degrees
for joint1 and joint2. The feedback gain matrices of the PD controller were determined

as
)1.0,6.0(diag
p
k
,
)3.0,8.0(diag
v
k
. Learning rates in (12) and (13) were chosen
as
07.0
1

j

,
04.0
2

j


)4.,1( j
,
)4.,1;4,,1(01.0   ji
ij

. Simulations were taken
place under Matlab environment.
Fig.7 ~ Fig.10 show an example of the simulations. Fig.7 gives the planned joint trajectories

and tracking control results. The broken lines indicate the planned trajectories which are not
easy to be seen since they are almost completely covered by the thick lines i.e. the tracking
results in fourth time learning. The dotted lines indicate results according PD control only,
and the thin lines are first learning results.
Fig.8 gives velocity tracking results with the lines’ types being the same meaning as
described for Fig.7. Fig.9 shows control inputs of joint 1, (b) and (c) are control input
generated by PD controller and neural network controller, respectively. (a) is the whole
control input, i.e. the summation of (b) and (c). Fig.10 shows control inputs of joint 2.
From the simulation results it is seen that using the combined control system with PD
controller and neural net work controller high precise joint trajectory tracking performance
can be achieved under learning process of the weights of the neural network.


Fig. 7. Simulation results: planned joint trajectories and tracking results.


Fig. 8. Simulation results: planned joint velocity trajectories and tracking results.


Fig. 9. Simulation results: control inputs of joint 1.


Fig. 10. Simulation results: control inputs of joint 2.
TrajectoryControlofRobotManipulatorsUsingaNeuralNetworkController 371

Focusing on control of the most complex part of the robot, we do not take the third joint into
consideration in the control design. The fourth joint is extremely light-weight designed
comparing with other joints and its link length is zero. Fourth joint does not cause dynamic
coupling to the others. Therefore, we do not take it into consideration as well in control
design and experiments.

6.2 Trajectory tracking control simulations
The joint trajectory tracking control simulations were carried out based on a simplified
dynamic model of (3). The neural network controller was designed with three layers, four
nodes for the input layer and hidden layer respectively, and two nodes for the output layer.
The learning scheme was designed using the method given in section IV. The desired joint
trajectories are designed using triangle functions with amplitudes to be 45 and 30 degrees
for joint1 and joint2. The feedback gain matrices of the PD controller were determined
as
)1.0,6.0(diag
p
k
,
)3.0,8.0(diag
v

k
. Learning rates in (12) and (13) were chosen
as
07.0
1

j

,
04.0
2

j


)4.,1( 

j
,
)4.,1;4,,1(01.0 



ji
ij

. Simulations were taken
place under Matlab environment.
Fig.7 ~ Fig.10 show an example of the simulations. Fig.7 gives the planned joint trajectories
and tracking control results. The broken lines indicate the planned trajectories which are not
easy to be seen since they are almost completely covered by the thick lines i.e. the tracking
results in fourth time learning. The dotted lines indicate results according PD control only,
and the thin lines are first learning results.
Fig.8 gives velocity tracking results with the lines’ types being the same meaning as
described for Fig.7. Fig.9 shows control inputs of joint 1, (b) and (c) are control input
generated by PD controller and neural network controller, respectively. (a) is the whole
control input, i.e. the summation of (b) and (c). Fig.10 shows control inputs of joint 2.
From the simulation results it is seen that using the combined control system with PD
controller and neural net work controller high precise joint trajectory tracking performance
can be achieved under learning process of the weights of the neural network.


Fig. 7. Simulation results: planned joint trajectories and tracking results.

Fig. 8. Simulation results: planned joint velocity trajectories and tracking results.


Fig. 9. Simulation results: control inputs of joint 1.


Fig. 10. Simulation results: control inputs of joint 2.
RobotManipulators,TrendsandDevelopment372

6.3 Trajectory tracking control experiments
The joint trajectory tracking control experiments were carried out under almost same
conditions of the simulation except the feedback gain matrices were chosen as
)6.0,5.1(diag
p
k
,
)4.0,3.1(diag
v
k
, and amplitudes of the trajectories of joint 1 and 2 are
planned as 25 and 20 degrees.
Fig.11~Fig.14 show the experimental results. Meaning of each figure stands for the same
corresponding to the simulation results shown in the last subsection, as well as the lines in
figures.
From the experimental results, it can be seen that though the trajectory tracking accuracy is a
little bit lower comparing with the simulation results, the trajectory tracking error becomes
less and less when learning time increases. It confirms the effectiveness and usefulness of
the proposed control method.

Fig. 11. Experimental results: planned joint trajectories and tracking results.


Fig. 12 Experimental results: planned joint velocity trajectories and tracking results.


Fig. 13. Experimental results: control inputs of joint 1.

Fig. 14. Experimental results: control inputs of joint 2.

6.4 Discussion
PID controller controlled robot system is essentially driven by position error or trajectory
tracking error. In dynamic trajectory tracking of a robot under PD control, the PD controller
plays two important roles: one is motion regulation for guaranteeing stability of the robot
system; another one is to generate force/torque required by the dynamic trajectory to drive
the robot such that it would follow the trajectory. The latter needs a big enough tracking
error in order to generate actuating force/torque required by the trajectory for the robot.
From the simulation results one can see that the tracking error significantly decreases as
learning time increases while the control inputs generated by the neural network controller.
Comparing Fig.9 (b) with (c) or Fig.10 (b) with (c) it can be interestingly found that learning
for four times the neural network controller took PD controller over and played the main
role in generating actuation voltages for the robot. On the other hand, in the experimental
TrajectoryControlofRobotManipulatorsUsingaNeuralNetworkController 373

6.3 Trajectory tracking control experiments
The joint trajectory tracking control experiments were carried out under almost same
conditions of the simulation except the feedback gain matrices were chosen as
)6.0,5.1(diag
p

k
,
)4.0,3.1(diag
v

k
, and amplitudes of the trajectories of joint 1 and 2 are
planned as 25 and 20 degrees.
Fig.11~Fig.14 show the experimental results. Meaning of each figure stands for the same
corresponding to the simulation results shown in the last subsection, as well as the lines in
figures.
From the experimental results, it can be seen that though the trajectory tracking accuracy is a
little bit lower comparing with the simulation results, the trajectory tracking error becomes
less and less when learning time increases. It confirms the effectiveness and usefulness of
the proposed control method.


Fig. 11. Experimental results: planned joint trajectories and tracking results.


Fig. 12 Experimental results: planned joint velocity trajectories and tracking results.


Fig. 13. Experimental results: control inputs of joint 1.

Fig. 14. Experimental results: control inputs of joint 2.

6.4 Discussion
PID controller controlled robot system is essentially driven by position error or trajectory
tracking error. In dynamic trajectory tracking of a robot under PD control, the PD controller

plays two important roles: one is motion regulation for guaranteeing stability of the robot
system; another one is to generate force/torque required by the dynamic trajectory to drive
the robot such that it would follow the trajectory. The latter needs a big enough tracking
error in order to generate actuating force/torque required by the trajectory for the robot.
From the simulation results one can see that the tracking error significantly decreases as
learning time increases while the control inputs generated by the neural network controller.
Comparing Fig.9 (b) with (c) or Fig.10 (b) with (c) it can be interestingly found that learning
for four times the neural network controller took PD controller over and played the main
role in generating actuation voltages for the robot. On the other hand, in the experimental
RobotManipulators,TrendsandDevelopment374

results Fig.13 and Fig.14 “the roles changing” seems not as evident as in their simulation
counterparts. The reason lies on a fact that we had added the dead-zone compensating
inputs into PD control inputs.
Standing on the dynamics point of view, with the same tracking accuracy to a designed
dynamic trajectory, the whole control input should be the same judging with the same unit
of the input (whatever it counted by torque/force or voltage) regardless what kind of
control method is adopted. In robot control with neural network, it is a popular way to use a
neural network to approximate dynamics of the robot rather than use it as a controller itself.
From the simulation and experimental results, it can be concluded that the neural network
controller proposed in this paper not only plays the role as a controller but also play the role
to generate force/torque required by dynamic trajectories just as an approximated dynamic
model using neural network in computed torque control.
Though the results given here are limited on 4
th
time learning for the simulation and 6
th
time
learning for the experiment, we carried out much more simulations and experiments,
learning for 20 times, for example. The results show that after some specified time the

learning effect will remain unchanged.

7. Conclusions

In this article, we presented dynamic trajectory tracking control of industrial robot
manipulators using a PD controller and a neural network controller. Some different kinds
strucutres of neural network control systems were discuessed. The neural network
controller was designed as a three layers feed-forward network. The learning law of weights
of the neural network was derived using a simplified dynamic model of the robot and back
propagation approach. Dynamic trajectory tracking control simulations and experiments
were carried out using an industrial manipulator AdeptOne XL robot. The results showed
the effectiveness and usefulness of the proposed control method. From the simulations and
experiments, it was seen that according the increase of learning times the neural network
controller took over of the PD controller on playing the role in generating actuating
force/torque required by the dynamic trajectory. It also was clarified that the learning effect
of the neural network has some limitation, i.e. after some specified time of learning,
trajectory tracking accuracy remains unchanged.

8. References

Hsia, T. C .S (1989). A new technique for robust control of servo systems,
IEEE Transactions on Industrial Electronics, Vol. 36, No.1, pp.1-7
Kou, C. Y.; Wang, S. P. (1989). Nonlinear robust industrial robot control,
Transactions ASME, Journal of Dynamic Systems, Measurement and Control, Vol.111,
No.1, pp 24-30
Slotine, J. J. E.; Li, W. P. (1989). Composite adaptive control of robot manipulators,
Automatica, Vol. 25, No. 4, p 509-519,
Spong, M. W. (1992). On the Robust Control of Robot Manipulators, IEEE Transactions on
Automatic Control, Vol. 37, no. 11, pp. 1782-1786

Cheah, C. C.; Liu C. & Slotine, J. J. E. (2006). Adaptive Tracking Control for Robots with
Uncertainties in Kinematic, Dynamic and Actuator Models, IEEE Transactions on
Automatic Control, Vol.51 No.6, pp.1024-1029
Kwan, C.; Lewis, F. L. (2000). Robust backstepping control of nonlinear systems using neural
networks, IEEE Transactions on Systems, Man, and Cybernetics Part A:Systems and
Humans., Vol. 30, No. 6, pp.753-766
Sanger, T. D. (1994). Neural network learning control of robot manipulators using gradually
increasing task difficulty, IEEE Transactions on Robotics and Automation, Vol. 10, No.
pp.323-333
Jung, S.; Yim, S. B. (2001). Experimental Studies of Neural Network Impedance Control for
Robot Manipulators, Proceedings of IEEE International Conference on Robotics and
Automation, pp.3453-3458
Yu, W. S.; Wang, G. C. (2001). Adaptive Control Design Using Delayed Dynamical Neural
Network for a Class of Nonlinear Systems, Proceedings of IEEE International
Conference on Robotics and Automation, pp.3447-3452
Kim, Y. H.; Lewis, F. L. (1999). Neural Network Output Feedback Control of Robot
Manipulators , IEEE Transactions on Robotics and Automation, Vol.15, No. 2, pp.301-
309
Jung, S.; Hsia, T. C. (1996). Neural Network Reference Compensation Technique for
Position Control of Robot Manipulator, IEEE Conference on Neural Network, pp 1735-
1741, June
Barambones, O.; Etxebarria, V. (2002). Robust Neural Control for Robotic Manipulators,
Automatica, Vol.38, pp.235-242
Clark, C. M.; Mills, J. K. (2000). Robotic System Sensitivity to Neural Network Learning
Rate: Theory, Simulation, and Experiments, The International Journal of Robotics
Research, Vol.19,No.10, pp.955-968
TrajectoryControlofRobotManipulatorsUsingaNeuralNetworkController 375

results Fig.13 and Fig.14 “the roles changing” seems not as evident as in their simulation
counterparts. The reason lies on a fact that we had added the dead-zone compensating

inputs into PD control inputs.
Standing on the dynamics point of view, with the same tracking accuracy to a designed
dynamic trajectory, the whole control input should be the same judging with the same unit
of the input (whatever it counted by torque/force or voltage) regardless what kind of
control method is adopted. In robot control with neural network, it is a popular way to use a
neural network to approximate dynamics of the robot rather than use it as a controller itself.
From the simulation and experimental results, it can be concluded that the neural network
controller proposed in this paper not only plays the role as a controller but also play the role
to generate force/torque required by dynamic trajectories just as an approximated dynamic
model using neural network in computed torque control.
Though the results given here are limited on 4
th
time learning for the simulation and 6
th
time
learning for the experiment, we carried out much more simulations and experiments,
learning for 20 times, for example. The results show that after some specified time the
learning effect will remain unchanged.

7. Conclusions

In this article, we presented dynamic trajectory tracking control of industrial robot
manipulators using a PD controller and a neural network controller. Some different kinds
strucutres of neural network control systems were discuessed. The neural network
controller was designed as a three layers feed-forward network. The learning law of weights
of the neural network was derived using a simplified dynamic model of the robot and back
propagation approach. Dynamic trajectory tracking control simulations and experiments
were carried out using an industrial manipulator AdeptOne XL robot. The results showed
the effectiveness and usefulness of the proposed control method. From the simulations and
experiments, it was seen that according the increase of learning times the neural network

controller took over of the PD controller on playing the role in generating actuating
force/torque required by the dynamic trajectory. It also was clarified that the learning effect
of the neural network has some limitation, i.e. after some specified time of learning,
trajectory tracking accuracy remains unchanged.

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